1、niiiar0u(u)In general,a degree-n polynomial curve can be used to fit(n+1)data points.P1P0t1t0r(u)r(u)=U A=U MV with 0 u 13232320011320011321 22113321100101000232+32 uuuuuuuuuPPPPuuuHVtttt with 0 u 1 00n!()()(1)i!(n i)!nnin-iii,niiiuBuu-urPPOn evaluating the Bezier equation and its derivative at u=0,
2、1 r(0)=V0r(1)=Vnr(0)=n(V1 V0)r(1)=n(Vn Vn-1)Bezier found a family of functions called Bernstein Polynomials that satisfy these conditions:V0V1V2V3V3V2V1V0V2V1V0V3r(u)=(1 u)3 V0+3u(1 u)2 V1+3u2(1 u)V2+u3 V3 r(u)=U M R 01322313313630133001000uuu VVVVr(0)=V0r(0)=3(V1 V0)r(1)=V3r(1)=3(V3 V2)The shape of
3、 the curve resembles that of the control polygon.with 0 u 1 0()()nii,niuNu rVNi,n(u)=n-i1j 01(1)C(u+n-i-j)!jjnnnThe primary functionB-spline Model defined by n+1 points Vi is given by the 1(n 1)!C!(n 1)!jnjj Wherer(t)=t2 t 1 =U3 M3 P3 0 t 1012121220110 VVV Cubic uniform B-spline model with control p
4、oints V0,V1,V2,and V3 r(t)=1/6 u3 u2 u 1 =U4 M4 P4 0 t 101231331363030301410VVVVTwo curve segments ra(u)and rb(u)q ra(1)=P1=rb(0)(C0-continuous)q ra(1)=t1=rb(0)(C1-continuous)q ra(1)=rb(0)(C2-continuous)q Collectively called a parametric C2-condition.q The composite curve to pass through P0,P1,P2,an
5、d the tangents t0 and t2 are assumed to be given.Thus,the problem here is to determine the unknown t1 so that the two curve segments are C2-continuous at the common join P1.P0P1P2t2t0t1=?ra(u)rb(u)C=1122123301000001Sa=P0 P1 t0 t1T Sb=P1 P2 t1 t2TApplying C2 continuity:ra(1)=6P0 6 P1+2t0+4t1rb(0)=-6P
6、1+6 P2-4t1-2t2C0-continuity and C1-continuityalready applied3300iji jijduv00010203101112132021222330313233dddddddddddddddd221133210010100011101110010001001110111001000100 xxssxxssttPPttPPr(u,v)=U M B MTVT 0 u,v 1 WhereM=B=The matrix M is called a(cubic)Bezier coefficient matrix,and B is called a Bezier control point net which forms a characteristic polyhedron.303033ijijjiV(v)B(u)B,133136303300100033323130232221201312111003020100VVVVVVVVVVVVVVVV33,3,300(u)(v)ijijijNNV 133136301303061410r(u,v)g(u)r(u,)External edge loopInterior edge loopn All of these form the scope of test in the final exam.
